We introduce a form of pre-play communication that we call « preopening ». During the preopening, players announce their tentative actions to be played in the underlying game. Announcements are made using a posting system which is subject to stochastic failures. Posted actions are publicly observable and players payoffs only depend on the opening outcome, i.e. the action profile that is posted at the end of the preopening phase. We show that when the posting failures hit players idiosyncratically all equilibria of the preopening game lead to the same opening outcome that corresponds to the most « sensible » pure Nash equilibrium of the underlying game. By contrast preopening does not operate an equilibrium selection when posting failure hits players simultaneously.

Keywords:

Preopening; equilibrium selection; bargaining; cheap talk

JEL:

C72

A classical algorithm to break through Maskin’s theorem for small-scale cases

Quantum mechanics has been applied to game theory for years. A recent work [H. Wu, Quantum mechanism helps agents combat « bad » social choice rules. \emph{International Journal of Quantum Information}, 2010 (accepted). Also see http://arxiv.org/pdf/1002.4294v3] has generalized quantum mechanics to the theory of mechanism design (a reverse problem of game theory). Although the quantum mechanism is theoretically feasible, agents cannot benefit from it immediately due to the restriction of current experimental technologies. In this paper, a classical algorithm is proposed to help agents combat « bad » social choice rules immediately. The algorithm works well when the number of agents is not very large (e.g., less than 20). Since this condition is acceptable for small-scale cases, it can be concluded that the Maskin’s sufficiency theorem has been broken through for small-scale cases just right now. In the future, when the ex perimental technologies for quantum information are commercially available, the Wu’s quantum mechanism will break through the Maskin’s sufficiency theorem completely.

We analyze cooperative behavior in a prisoner’s dilemma game with high stakes, face-to-face communication, and two rounds of pre-play in which the two final contestants are endogenously selected via a voting process. Using data from the British television game show « Golden Balls », we find a unilateral cooperation rate of 55% and a mutual cooperation rate of 33%. The stake size is on average $13,300 and ranges from $3 to $100,150. Our analysis shows that both stake size and communication have a significant impact on the player’s likelihood to cooperate. In particular, we observe a negative correlation between stake size and cooperation. Also certain gestures, as handshakes, decrease the likelihood to cooperate. But, if players mutually promise each other to cooperate and in addition shake hands on it, the cooperation rate increases. We also show that a player’s expectation about the stake size matters. Further, we find a s trong link between contestant’s pre-play behavior and the outcome of the prisoner’s dilemma. Players who contribute more to the stake size are less likely to cooperate, even though each player’s contribution is determined by a random process. Apart from that, it matters whether a player has lied in the pre-play and whether she experienced her opponent’s goodwill. Addressing the partner selection process, we find that contestant’s voting decisions are based on objective criteria, i.e., their opponent’s monetary contribution to the stake size, as well as subjective personal characteristics, i.e., the opponents’ trustworthiness.

In the context of a “beauty contest” coordination game (in which payoffs depend on the quadratic distance of actions from an unobserved state variable and from the average action) players choose how much costly attention to pay to various informative signals. Each signal has an underlying accuracy (how precisely it identifies the state) and a clarity (how easy it is to understand). The unique linear equilibrium has interesting properties: the signals which receive attention are the clearest available, even if they have poor underlying accuracy; the number of signals observed falls as the complementarity of players’ actions rises; and, if actions are more complementary, the information endogenously acquired in equilibrium is more public in nature. The consequences of “rational” inattention constraints on information transmission and processing are also studied.

It is well-known that there are no pure strategy Nash equilibria (PSNE) in the standard three-candidate spatial voting model when candidates maximize their share of the vote. When all that matters to the candidates is winning the election, however, we show that PSNE do exist. We provide a complete characterization of such equilibria and then extend our results to elections with an arbitrary number of candidates. Finally, when two candidates face the potential entrant of a third, we show that PSNE no longer exist, however, they do exist when the number of existing candidates is at least three.

Keywords:

Voting, spatial equilibrium, location models, entry.

JEL:

C7

Testing the TASP: An Experimental Investigation of Learning in Games with Unstable Equilibria

We report experiments designed to test between Nash equilibria that are stable and unstable under learning. The “TASP” (Time Average of the Shapley Polygon) gives a precise prediction about what happens when there is divergence from equilibrium under a wide class of learning processes. We study two versions of Rock-Paper-Scissors with the addition of a fourth strategy, Dumb. The unique Nash equilibrium places a weight of 1/2 on Dumb in both games, but in one game the NE is stable, while in the other game the NE is unstable and the TASP places zero weight on Dumb. Consistent with TASP, we find that the frequency of Dumb is lower and play is further from Nash in the high payoff unstable treatment than in the other treatments. However, the frequency of Dumb is substantially greater than zero in all treatments.

There is no consensus on the properties of voting power indices when there are a large number of voters in a weighted voting body. On the one hand, in some real-world cases that have been studied the power indices have been found to be nearly proportional to the weights (eg the EUCM, US Electoral College). This is true for both the PenroseBanzhaf and the Shapley-Shubik indices. It has been suggested that this is a manifestation of a conjecture by Penrose (known subsequently as the Penrose limit theorem, that has been shown to hold under certain conditions). On the other hand, we have the older literature from cooperative game theory, due to Shapley and his collaborators, showing that, where there are a nite number of voters whose weights remain constant in relative terms, and where the quota remains constant in relative terms, while the total number of voters increases without limit – so called oceanic games – the powers of the voters with nite weight tend to limiting values that are, in general, not proportional to the weights. These results, too, are supported by empirical studies of large voting bodies (eg. the IMF/WB boards, corporate shareholder control). This paper proposes a restatement of the Penrose Limit theorem and shows that, for both the power indices, convergence occurs in general, in the limit as the Laakso-Taagepera index of political fragmentation increases. This new version reconciles the di erent theoretical and empirical results that have been found for large voting bodies

Communication, Commitment, and Deception in Social Dilemmas: Experimental Evidence

Social norms of cooperation are studied under several forms of communication. In an experiment, strangers could make public statements before playing a prisoner’s dilemma. The interaction was repeated indefinitely, which generated multiple equilibria. Communication could be used as a tool to either signal intentions to coordinate on Pareto-superior outcomes, to deceive others, or to credibly commit to actions. Some forms of communication did not promote the incidence of efficient Nash play, and sometimes reduced it. Surprisingly, cooperation suffered when subjects could publicly commit to actions.

We study cooperation in four-person economies of indefinite duration. Subjects interact anonymously playing a prisoner’s dilemma. We identify and characterize the strategies employed at the aggregate and at the individual level. We find that (i) grim trigger well describes aggregate play, but not individual play; (ii) individual behavior is persistently heterogeneous; (iii) coordination on cooperative strategies does not improve with experience; (iv) systematic defection does not crowd-out systematic cooperation.

Building on Battigalli and Dufwenberg (2009)’s framework of dynamic psychological games and the recent progress in the modeling of dynamic unawareness, we provide a general framework that allows for `unawareness’ in the strategic interaction of players motivated by belief-dependent psychological preferences like reciprocity and guilt. We show that unawareness has a pervasive impact on the strategic interaction of psychologically motivated players. Intuitively, unawareness influences players’ beliefs concerning, for example, the intentions and expectations of others which in turn impacts their behavior. Moreover, we highlight the strategic role of communication concerning feasible paths of play in these environments.

How and why do groups form? In many cases, group formation is endogenous to the actions that individual members take and the norms associated with these actions. In this paper, we conduct an experiment that allows groups to form endogenously in the context of the classic voluntary contribution mechanism public goods game. We identify unproductive costs – “sacrifice” – as a mechanism for endogenous group formation, a result which is consistent with the “sacrifice and stigma” theory of religious groups. We find that changes in relative prices (between private and public goods) act to screen out free-riders, subjects who choose high-sacrifice groups contribute more to the public good once in these groups, and moderate welfare gains are available to those who voluntarily incur unproductive costs.

This experiment compares the performance of two contest designs: a standard winnertake- all tournament with a single fixed prize, and a novel proportional-payment design in which that same prize is divided among contestants by their share of total achievement. We find that proportional prizes elicit more entry and more total achievement than the winner-take-all tournament. The proportional-prize contest performs better by limiting the degree to which heterogeneity among contestants discourages weaker entrants, without altering the performance of stronger entrants. These findings could inform the design of contests for technological and other improvements, which are widely used by governments and philanthropic donors to elicit more effort on targeted economic and technological development activities.

This paper examines conflicts in which performance is measured by the players’ success or failure in multiple component conflicts, commonly termed « battlefields. » In multi-battlefield conflicts, behavioral linkages across battlefields depend both on the technologies of conflict within each battlefield and the nature of economies or diseconomies in how battlefield out- comes and costs aggregate in determining payos in the overall conflict.

We develop a competitive equilibrium theory of a market for votes. Before voting on a binary issue, individuals may buy and sell their votes with each other. We define the concept of Ex Ante Vote-Trading Equilibrium, identify weak sufficient conditions for existence, and construct one such equilibrium. We show that this equilibrium must always result in dictatorship and the market generates welfare losses, relative to simple majority voting, if the committee is large enough. We test the theoretical implications by implementing a competitive vote market in the laboratory using a continuous open-book multi-unit double auction.